Cash-in-advance constraints, bubbles and
monetary policy
∗
Philippe Michel
†Bertrand Wigniolle
‡March 11, 2004
∗The authors would like to thank Enrico Minelli for his helpful comments and
sugges-tions. They are also indebted to two anonymous referee for very relevant advice.
†GREQAM, Université de la Méditerranée II and EUREQua. GREQAM, Centre de la
Vieille Charité, 2 rue de la Charité, 13002 Marseille, France.
‡EUREQua Université de Paris I and L.I.B.R.E. Université de Franche-Comté.
Ad-dress: EUREQua, Université de Paris-I, Maison des Sciences Economiques, 106-112, boule-vard de l’hôpital, 75647 Paris Cedex 13, France. Email : wignioll@univ-paris1.fr. Tel : 33+ (0)1 44 07 81 98, Fax : 33+ (0)1 44 07 82 31
halshs-00268861, version 1 - 1 Apr 2008
Author manuscript, published in "Macroeconomic Dynamics 9, 1 (2005) 28-56" DOI : 10.1017/S1365100505040113
Abstract
In this paper, we study the equilibrium dynamics of an overlapping gen-erations model with capital, money and cash-in-advance constraints. At each period, the economy can experience two different regimes: either the cash-in-advance constraint is binding and money is a dominated asset, or the con-straint is strictly satisfied and money has the same return as capital. When the second regime occurs, we say that the economy experiences a temporary bubble. We show the existence of temporary bubbles, and we prove that cyclical equilibria may exist. In these equilibria, the economy experiences some periods without bubbles and some periods with bubbles. We also show that monetary creation can be used in order to eliminate temporary bubbles.
JEL numbers: D9, E5 and G1.
Key words: overlapping generations model, bubbles, cash-in-advance con-straint, monetary policy.
1
Introduction
The interplay between monetary policy and macroeconomicfluctuations is a standard theme in monetary analysis, including, among many authors, the famous contributions of Keynes (1964) and Friedman (1969). Recent theoret-ical works consider this question within general equilibrium frameworks, and particularly within the overlapping generations model. Endogenous volatil-ity arises from the existence of multiple equilibria and indeterminacy, whose existence may depend on monetary policy.
Samuelson (1958) and Tirole (1985) have greatly contributed to the pio-neering work in this field. Tirole presents a benchmark model where money is viewed as a rational bubble, which is valued only if its return is equal to the rate of return on capital. He shows that real money balances held by the agents can only be valued in an economy that is inefficient without money. In this case, an infinity of equilibria exist: one of these equilibria leads to a constant and positive value in the long run for the bubble held by each agent, and the economy converges towards the golden rule; the other trajec-tories converge to the stationary state of Diamond’s model without bubbles. Tirole also shows that the introduction of a reserve requirement constraint precludes the existence of asymptotic trajectories without bubbles.
Hahn and Solow (1995) study a framework close to Tirole (1985). They consider the standard overlapping generation frameworkà laDiamond (1965), in which money is introduced by a cash-in-advance constraint. They study the intertemporal equilibrium under the assumption that the liquidity con-straint is binding in each period: the rate of return for money must be smaller than the rate of return for capital.
More recent literature on monetary analysis explicitly models credit mar-ket frictions and financial intermediaries. Bhattacharya, Guzman, Huybens and Smith (1997) and Schreft and Smith (1998) introduce spatial separa-tion and limited communicasepara-tion between agents. These assumpsepara-tions provide micro-foundations for money holding: money is held even if it is a dominated asset. These authors also assume that agents are subject to stochastic re-locations that act like shocks to their portfolio preferences, thereby creating an explicit role for the banking system. The main findings of these studies are obtained from analyzing a situation where the government issues both money and bonds. In Schreft and Smith (1998), monetary creation allows the government to pay back interest payments on public debt. In Bhattacharya, Guzman, Huybens and Smith (1997), money also makes it possible tofinance the current government deficit. Both contributions show that these policies
can lead to the existence of many steady-state equilibria and indeterminacy1.
Boyd and Smith (1998) and Huybens and Smith (1998) introduce another form of credit market. They assume that only the project owner can observe at no cost the return on an investment project. Such a framework allows for the existence of credit rationing and gives an explicit allocative function to
financial intermediation. This leads to the existence of multiple equilibria, whose existence may depend on the rate of monetary growth.
Michel and Wigniolle (2003) bring a new argument to this literature in a simple example, which is based on Hahn and Solow’s (1995) model. Hahn and Solow (1995), Bhattacharya, Guzman, Huybens and Smith (1997), Schreft and Smith (1998) and Gomis-Porqueras (2000) focus on equilibria where money is a dominated asset. On the contrary, in Tirole (1985), Boyd and Smith (1998) and Huybens and Smith (1998), money and bank deposits have the same rate of return. Therefore the literature has only focused on the case of permanent regimes, in which money is either dominated or not dominated at all dates. In contrast to this literature, Michel and Wigniolle (2003) consider that along an intertemporal equilibrium, the economy can experience both periods where money and capital have the same return (“Tirole’s regime”) and periods where money is a dominated asset (“Hahn and Solow’s regime”). Periods in which Tirole-regime occurs are called tem-porary bubbles. Such temtem-porary bubbles may exist in an economy, which would experience under-accumulation without money, when the weight of the cash-in-advance constraint is not too hard.
In this paper we propose a twofold generalization of Michel and Wigniolle (2003). Firstly we deal with general formulations for preferences and pro-duction function. Secondly, we introduce monetary policy. Then, we can prove that the existence of temporary bubbles is a general property that can arise as soon as the economy without money (the economy corresponding to Diamond’s (1965) model) have a stationary state associated with under-accumulation, and when the weight of the cash-in-advance constraint is not too hard. Moreover, if the stationary state of the economy without money is associated with over-accumulation, there exists a large range of rate of mon-etary creation such that temporary bubbles exist. These results are reached by doing a local study within the neighborhood of an stationary equilibrium. Is is proved that cyclical equilibria may exist where the economy experiences nperiods of temporary bubbles andpperiods without bubbles,nandpbeing some integers. Thus, a multiplicity of equilibria exists.
The influence of monetary creation is another new aspect that we consider 1As stressed by Gomis-Porqueras (2000), the existence of multiple steady states in these
models depends on the design of the monetary policy.
in our framework. What is the impact of monetary policy on temporary bubbles? Intuition suggests that monetary creation induces inflation, which causes a drop in the return of money detention. In that case, the return on money could no longer be equal to the return on capital. In the particular case of Cobb-Douglas functions for utility and production, we show that a high enough rate of monetary creation eliminates temporary bubbles. Thus, the economy follows the only equilibrium trajectory without bubbles and indeterminacy vanishes.
Monetary creation can be viewed as a counter-bubble weapon capable of stabilizing the economy on the bubbleless equilibrium. However, such a policy is not Pareto improving. Indeed, a one period bubble is beneficial for the generation living during the bubble period, even if it is detrimental for the following generations. Moreover, monetary creation increases the distortion between the returns on money and capital savings. But we develop two arguments in favor of such a monetary policy. For each argument, we compare a trajectory where monetary creation have eliminated bubbles with a trajectory with bubbles. Firstly, considering the long run, we show that there exist parameter values of the model such that the long run utility levels are smaller along a trajectory with the periodic appearance of bubbles than along the bubbleless trajectory. Secondly, considering the short run, we show that for any integer P,there exist parameter values such that the only appearance of a one period bubble is enough to drop the utility levels during P periods under the level reached along the bubbleless trajectory. For these two reasons, fighting bubbles can be an objective of monetary policy.
In our model, indeterminacy and multiple equilibria are the by-products of an economy with money and capital. A temporary bubble on money can be interpreted as a deflation period, where money and capital have the same return. It creates a drop in capital accumulation and production because it absorbs a share of savings. A monetary policy can eliminate such deflation periods as it precludes the appearance of bubbles. But these “Keynesian” features of monetary policy are not obtained by Keynesian mechanisms such as an increase in demand, but by bubbles elimination. Our results seems to contradict the conventional wisdom, which consider that an expansionist monetary policy can promote the appearance of bubbles. In fact, this is only an outward discrepancy as we are interested in bubbles on money, when the usual argument takes into account bubbles on asset prices. Our model shows that a too restrictive monetary policy can induce a deflationist bubble.
The paper is organized in the following way. The model is presented in the second section. The study of the intertemporal equilibrium is achieved
in section three. The fourth section studies the dynamics of the economy. Finally, a fifth section considers the counter-bubble monetary policy.
2
The model
We consider a standard overlapping generations model à la Allais (1947)-Diamond (1965), in which money is introduced by a cash-in-advanced con-straint. Agents live two periods. They supply one unit of labor in the first period (when young), and they are retired and consume the proceeds of their savings in the second period (when old). The number of young agents at a date t, Nt, grows at the constant raten:Nt= (1 +n)Nt−1.
2.1
Money and monetary policy
Following Hahn and Solow (1995), we assume that agents are subject to a cash-in-advance constraint: a shareµof consumption expenses in the second period must be financed by the amount of money saved during the first period,
Mt≥µPt+1dt+1 (1)
µ is a parameter such that 0 < µ < 1, dt+1 is the second period amount of
consumption and Pt+1 is the price of the good in money in periodt+ 1.
The government creates money, and it gives this money as a lump sum transfer to young people. We denote the total supply of money in period t by Mt and the rate of money creation byλt. Thus, we have:
Mt= (1 +λt)Mt−1 (2)
Tt is the lump sum transfer received by each of the Nt young agents living during period t. This transfer isfinanced by money creation:
λtMt−1 =NtTt (3)
2.2
The agents
Agents born in t are endowed with an intertemporal utility function:
Ut=U(ct, dt+1) (4)
ctis thefirst period consumption, anddt+1 is the second period consumption.
Assumption 1: U is strictly quasi-concave, twice continuously diff eren-tiable, and satisfies the Inada conditions. It is also assumed that candd are normal goods.
From this assumption, we can deduce the existence of a continuously differentiable functionσ(w, R) defined on R2
++ by: σ(w, R) =arg max σ U(w−σ, Rσ) (5) and that 0<σ0 1 <1.
Agents can invest their savings in capital and in money. In real terms, the two budget constraints of a generation t agent are:
ct+st+ Mt Pt = wt+θt (6) dt+1 = Rt+1st+ Mt Pt+1 (7) wherewtis the real wage in periodt andst is the amount of savings invested in capital. Rt+1 is the real return factor expected for period t+ 1. Mt is the money amount held in period t andPtis the price of the good in money.
θt=Tt/Pt is the real value of the money transfer.
2.3
Agents’ behavior
Each young agent born in period t maximizes his utility given by (4) under the budget constraints (6) and (7), and the liquidity constraint (1). As in Michel and Wigniolle (2003), we must distinguish at each period t the two possible cases:
The Hahn and Solow’s case (HS-regime): the liquidity constraint is binding: Mt/Pt+1 = µdt+1 and then the expected return on money is no
greater than the return on financial savings Pt/Pt+1 ≤ Rt+1. In that case,
using (6), (7) and the liquidity constraint to eliminate Mt andst, we obtain the intertemporal constraint:
ct+ dt+1 ρt+1 = wt+θt (8) with 1 ρt+1 = 1−µ Rt+1 +µPt+1 Pt (9)
ρt+1 is the real expected return of total savings when the liquidity constraint
is binding. 1/ρt+1 is the mean of the inverse return of money weighted by µ and the inverse return of capital weighted by 1−µ.
The resolution of the consumer program leads to the expression of total savings:
σt=σ(wt+θt,ρt+1) =st+mt (10) wheremt is the real money holding: mt ≡Mt/Pt.Using (7) and the liquidity constraint, we obtain:
(1−µ)mt=µ Pt+1
Pt
Rt+1st (11)
Equations (10) and (11) givest andmt. Finally, by using (11), the condition (9) can be replaced by:
ρt+1 =
Rt+1st
(1−µ)(mt+st)
(12) The three conditions (10), (11) and (12) characterize the behavior of a gen-eration t agent who is expecting a binding liquidity constraint.
The Tirole’s Case (T-regime): the liquidity constraint is not binding: Mt/Pt+1 > µdt+1 and the expected return on money must be equal to the
return on financial savings: Pt/Pt+1 = Rt+1. In this case, the consumer’s
total savings is the same as in the Diamond’s model:
σt=σ(wt+θt, Rt+1) =st+mt (13) The consumer’s savings can be shared by any proportion of money or capital. The only constraint is the cash-in-advance constraint, which is equivalent to:
(1−µ)mt > µst (14)
2.4
Firms
We assume that at each period t, there exists one competitive firm that uses neoclassical technology with constant returns to scale F(Kt, Lt). F is increasing in its two arguments, concave, twice continuously differentiable. Lt is the quantity of labor used in production, paid by the real wagewt. The profit maximization of the firm gives:
wt = FL(Kt, Lt) Rt = FK(Kt, Lt)
3
Intertemporal equilibrium
3.1
Equilibrium characterization
Defining the variable kt as Kt/Nt, the equilibrium wage and the factor of return for productive investments are:
wt = FL(kt,1)≡w(kt) (15)
Rt = FK(kt,1)≡R(kt) (16)
Mt being the total stock of money, we assume that it is held in equal shares by the agents. Equilibrium in the money market gives:
Mt=NtMt=NtPtmt (17)
which deduces the real value of the lump sum monetary transfer:
θt= λtMt−1 NtPt = λt 1 +λt mt (18)
Finally, we express that capital in period t+ 1 results from savings of generation t agents:
Kt+1=Ntst⇔(1 +n)kt+1 =st (19) Using (17), the money gross return is given by:
Pt/Pt+1 = Mt/(Ntmt) Mt+1/(Nt+1mt+1) = (1 +n)mt+1 (1 +λt+1)mt (20) This return cannot be larger than the return of physical capital, or, with kt+1 >0:
mt+1 ≤
Rt+1(1 +λt+1)
1 +n mt (21) The two preceding cases will be studied separately.
The HS-regime: the liquidity constraint is binding between t and t+ 1, and Pt/Pt+1 ≤Rt+1. Using the expression of the cash-in-advance constraint
(11), and equations (19) and (20), we obtain:
(1−µ)mt+1 =µ(1 +λt+1)Rt+1kt+1 (22)
The dynamics of capital is given by replacing savingsstin (19) by its expres-sion (13), and in using ρt+1 given by (12) and θt given by (18):
(1 +n)kt+1 = σ µ wt+ λt 1 +λt mt,ρt+1 ¶ −mt (23) withρt+1 = (1 +n)Rt+1kt+1 (1−µ)(mt+ (1 +n)kt+1)
Finally, the condition, which ensures that money is a dominated asset, cor-responds to equation (21). With (22), we can write:
µ(1 +n)kt+1 ≤(1−µ)mt (24)
The T-regime: money is not dominated between t and t+ 1: Pt/Pt+1 =
Rt+1. In this case, (21) is verified with an equality, and we have:
mt+1 =
Rt+1(1 +λt+1)
1 +n mt (25) The capital dynamics is always given by (23), but with ρt+1 =Rt+1 :
(1 +n)kt+1 =σ µ wt+ λt 1 +λt mt, Rt+1 ¶ −mt (26)
Finally, we must write that the cash-in-advance constraint of generation t agents is not binding, or (14). Using (19), we find:
µ(1 +n)kt+1 <(1−µ)mt (27)
Here, we recognize the same relation as (24), but written with a strict in-equality.
Finally, we have to make precise the equilibrium conditions for the first old agents in period t= 0. Their budget constraint is: d0 =M−1/P0+R0s−1
withs−1 =K0/N−1.They also must satisfy the cash-in-advance requirement:
M−1 ≥µP0d0. Thus, they must hold a real amount of money m0 such that:
(1−µ) m0 1 +λ0 ≥
µR0k0(1 +n) (28)
It is now possible to give the following definition:
Definition 1 : Given an initial value k0 and a sequence (λt)t≥0 of rates of
money creation, a sequence (kt, mt)t≥0 with kt > 0 and mt > 0 that satisfy
equations (15), (16) and (28), and for all t≥0,
• either the equilibrium conditions of the HS-regime: (22), (23) and (24)
• or the equilibrium conditions of the T-regime: (25), (26) and (27)
defines an intertemporal equilibrium with perfect foresight.
3.2
Two types of equilibrium dynamics
3.2.1 Dynamics in the HS-regime
Assuming that the liquidity constraint is binding along all the dynamics, from (22) and (23) these dynamics satisfy:
g(kt+1, kt, µ,λt) = 0 (29) with g(kt+1, kt, µ,λt) ≡ (1 +n)kt+1−σ µ wt+ λtµR(kt)kt 1−µ ,η(kt+1, kt, µ,λt) ¶ +µ(1 +λt)R(kt)kt 1−µ andη(kt+1, kt, µ,λt) ≡ (1 +n)R(kt+1)kt+1 µ(1 +λt)R(kt)kt+ (1−µ)(1 +n)kt+1
This is a one-dimensional dynamics of kt.
Assuming λ is constant, a stationary state in the HS-regime (k∗, m∗)
satisfies the following equations:
(1 +n)k∗+ µ(1 +λ) 1−µ R(k ∗)k∗ = σ Ã w(k∗), 1 µ1+1+λn +R1(−kµ∗) ! m∗ = µ(1 +λ) 1−µ R(k ∗)k∗
From (24), (k∗, m∗) are such that money is a dominated asset if: R(k∗) ≥
(1 +n)/(1 +λ).
3.2.2 Dynamics in the T-regime
In the T-regime, equations (25) and (26) define a dynamical system of dimen-sion 2, which does not depend on the cash-in-advance constraint. Providing that this constraint (condition (27)) is satisfied, the dynamics are the same as in the Diamond’s model with a bubble. This is the Tirole (1985) model with monetary creation (in the Tirole model, the stock of money is constant). The cash-in-advance constraint compels the money stock to retain a positive value. Capital is a pre-determined variable when the real value of money is a forward-looking variable.
Assuming a constant rate of money creation: λt=λ∀t,a stationary state
(k∗, m∗) must satisfy (25), or
R(k∗) = 1 +n
1 +λ (30)
equation (26), (1 +n)k∗ =σ µ w(k∗) + λ 1 +λm ∗,1 +n 1 +λ ¶ −m∗ (31)
and the constraint (27):
m∗ > µ(1 +n)
1−µ k
∗
From (30), the stationary state value k∗
T(λ) is an increasing function of λ, which is equal to the golden rule when λ= 0. Defining the function:
φλ(m) =σ µ w(k∗) + λ 1 +λm, 1 +n 1 +λ ¶ −m−(1 +n)k∗
φλ(m) is a decreasing function of m as:
φ0λ(m) = λ 1 +λσ
0
1−1<0
which becomes negative when m is high enough. Thus, a stationary state in the T-regime exists if:
φλ µ µ(1 +n) 1−µ k ∗ T(λ) ¶ >0
and, when it exists, it is unique.
4
Study of the dynamics
4.1
Local study of the dynamics in the HS-regime
We make a local study of convergent trajectories in the HS-regime for a small value of µ. We consider a constant rate of monetary creation λ and a stationary equilibrium of the Diamond’s economy kD, assumed to be stable, and which satisfies R¡kD¢>(1 +n)/(1 +λ). Forλ = 0, this last condition is equivalent to under-accumulation. If λ >0, the condition is weaker than under-accumulation. More precisely, for any value ofkD, there always exists a value of λ such this assumption 2 is satisfied.
Assumption 2: kDis a solution of(1 +n)kD=σ(w(kD), R(kD)), such that:
R(kD)> 1 +n 1 +λ and σ 0 ww0(k D) +σ0 RR0(k D)<1 +n
The last condition is equivalent to: dkt+1 dkt = σ 0 ww0(kD) 1 +n−σ0 RR0(kD) strictly between 0 and 1.
This assumption means that a steady state of the Diamond’s economy exists, which satisfiesR¡kD¢>(1 +n)/(1 +λ). Furthermore, the dynamics in a neighborhood of this steady state are monotonic and convergent.
Proposition 1 : There exists a neighborhood I = (k, k) of kD and ε > 0,
such that, for allµ∈(0,ε)and allk0 ∈I,there exists a unique intertemporal
equilibrium (kt, mt)t≥0 of the economy with a liquidity constraint, with initial
conditions k0 and m0 = µ(1+1−µλ)R0k0. This equilibrium is located in I (i.e.
kt ∈I ∀t),and such that at each date the liquidity constraint is binding
(HS-regime). This trajectory converges towards a stationary equilibrium k∗(µ) of
the HS-regime. Moreover, the sequence kt(µ) converges uniformly towards
the Diamond trajectory kt(0) starting from k0 when µ tends to 0.
Proof: see Appendix 1.
This proposition shows that within a neighborhood of a stationary state that satisfies assumption 2, it is possible to define an intertemporal equilib-rium with money and binding liquidity constraints at each period, whenµis small enough. The trajectory of the monetary economy converges uniformly towards the non-monetary economy as µ tends to 0. We will see, however, that this intertemporal equilibrium in the HS-regime is not unique, and that other equilibria with temporary bubbles exist.
4.2
Trajectories with bubbles
4.2.1 A one-period bubble
We will now study the following question: is it possible that a trajectory that converges in the HS-regime in the long run includes bubbles at certain dates ? Let us consider an intertemporal equilibrium, which is entirely in the HS-regime, as defined in proposition 1. Is it possible to modify this trajectory in one point in order to obtain a bubble (and to experience one period in the T-regime), and then to go back to the HS-regime?
A trajectory with one period in the T-regime can be characterized by the following conditions:
1. In t = 0, with k0 and m0 = µ(1+1−µλ)R0k0, there is no expected bubble;
the HS-regime occurs. Thus, we have: g(k1, k0, µ,λ) = 0
m1 =
µ(1 +λ) 1−µ R1k1
2. Int= 1,there is an expected bubble, and the economy experiences one period in the T-regime.
(1 +n)k2 =σ µ w1+ λ 1 +λm1, R2 ¶ −m1 m2 = R2(1 +λ) 1 +n m1 The liquidity constraint can be written as:
(1−µ)m1 > µ(1 +n)k2
or equivalently:
(1 +λ)R1k1 >(1 +n)k2
3. Int= 2, there is no expected bubble: we go back to the HS-regime.
(1 +n)k3 =σ µ w2+ λ 1 +λm2,ρ3 ¶ −m2 with ρ3 = (1 +n)R3k3 (1−µ)(m2+ (1 +n)k3) m3 = µ(1 +λ) 1−µ R3k3
4. After that, the dynamics of the HS-regime apply: fort ≥3, g(kt+1, kt, µ,λ) = 0
mt+1 =
µ(1 +λ)
1−µ Rt+1kt+1
Proposition 2 : Under Assumption 2, we can modify a trajectory of the
HS-regime sufficiently close to kD and for µ small enough in introducing a
bubble during one period.
Proof: In a stable Diamond’s equilibrium, which satisfies assumption 2 and in a neighborhood of this equilibrium, all these conditions are satisfied with µ= 0 as the inequality(1 +λ)R1k1 >(1 +n)k2. The implicit function
theorem makes it possible to determine k1, m1, k2, m2, k3 as functions of
µ, when µ is small enough, and k0 is sufficiently close to kD, these
func-tions satisfying condifunc-tions 1, 2 and 3. By continuity, it is possible to use neighborhoods, such that k3 ∈I. Proposition 1 can be applied fromk3.
Following the same argument, we alsofind that bubbles can be introduced during afinite number of periods. It is even possible to introduce bubbles at an infinite number of periods, which prevents the trajectory from converging towards a stationary state. To do this, all that is necessary is to introduce a bubble every time k is close enough to the stationary state of the dynamics without bubbles.
4.2.2 Cyclical bubbles
We will now consider a trajectory, along which the economy experiences an infinite number of periods with bubbles. We are looking for cyclical equilibria where the economy experiences n periods in the T-regime, and p periods in the HS-regime. Let us consider such a cycle of periodn+p.All the dynamics can be described by the orbit:
· µ m1 k1 ¶ µ m2 k2 ¶ ... µ mn kn ¶ µ mn+1 kn+1 ¶ µ mn+2 kn+2 ¶ ... µ mn+p kn+p ¶ ¸
Between periods1andn+1,the economy is in the T-regime; between periods n+ 1 and(n+p) + 1, the economy is in the T-regime, ...
In Appendix 2, we prove the following result:
Proposition 3 : Under assumption 2, ∀n and p, such that n ≥ 1 and
p≥1, for µ,when it is small enough, there exists a cycle of period n+p in the neighborhood of kD, such that the economy experiences n periods in the
T-regime, and p periods in the HS-regime.
The proof given in appendix 2 consists in writing the system of2 (n+p)
equations, which defines the cyclical orbit. These equations are satisfied for µ= 0 with ki =kD and mi = 0. Thus, the implicit function theorem makes it possible to show the existence of a solution
µ mi ki ¶ 1≤i≤n+p as a function of µ, when µis within a neighborhood ofµ= 0.Finally, it is possible to prove that this solution satisfies the constraints (24) and (27).
The result is obtained for µ small enough. Michel and Wigniolle (2002) study a Cobb-Douglas example without monetary creation, which makes it possible to characterize the global dynamics of the economy. They explicitly
find a decreasing relationship between the admissible period of the cycle (more precisely the admissible value of n) and the limit value forµ.
5
The Counter-bubble monetary policy
We will now consider the following question: is it possible to rule out tempo-rary bubbles with an appropriate monetary policy? The intuition suggests that monetary creation induces inflation, which causes a drop in the return of money detention. Thus, the return on money can no longer be the same as the return on capital. When a bubble appears, it absorbs a share of savings, which can no longerfinance productive investments. The bubble reduces cap-ital accumulation. Assuming the economy is in under-accumulation,fighting the possible appearance of bubbles could be an objective of the monetary policy. Another argument could be that the existence of temporary bubbles creates multiple equilibria and indeterminacy. If monetary creation pre-cludes the appearance of bubbles, it can stabilize the economy in the only remaining equilibrium in the HS-regime.
In this part, wefirst prove that an appropriate monetary policy can elim-inate bubbles. Then, we explain why a government which cares about agents welfare can wish to eliminate bubbles. In all the study, we now consider Cobb-Douglas functions for the utility and production functions.
5.1
The appropriate policy
The utility and production functions are now respectively given by:
U(ct, dt+1) = (1−a) lnct+alndt+1 (32)
F(Kt, Lt) = KtαL1t−α (33)
Under the assumption of a constant money stock, Michel and Wigniolle (2002) show that this formulation makes it possible to obtain an explicit form of the global dynamics followed by the economy.
Let us assume that the rate of money creation is constant: λt=λ,∀t.We introduce the new variable:
zt = wt mt
Appendix 3 shows that the dynamics of the economy can be summarized as:
HS-regime between t andt+ 1 :
zt+1 = 1−µ µ 1−α α 1 1 +λ ≡ze(λ) (34) zt ≤ 1 a µ 1 µ− aλ 1 +λ ¶ ≡z(λ) (35)
T-regime between t and t+ 1 :
zt+1 = (1−α)a α(1 +λ) · zt− µ 1 a − λ 1 +λ ¶¸ ≡φλ(zt) (36) zt < 1 a µ 1 µ − aλ 1 +λ ¶ =z(λ) (37)
and for the initial condition: z0 ≤ 1 1 +λ 1−µ µ 1−α α =ez(λ) (38)
In both regimes, the dynamics of(kt, mt) are given by:
(1 +n)kt+1 = kα t zt α(1 +λ)φλ(zt) (39) mt = (1−α)kα t zt (40) From (39), zt must satisfy another constraint to ensure that kt remains pos-itive in all periods: for all t,
φλ(zt)>0 (41)
We assume that the economy without money (µ = 0 and λ = 0) is in under-accumulation:
a(1−α)
α <1
The dynamics of this economy is similar to Michel and Wigniolle (2003) and can be summarized by the following diagram.
z 1/a φ0(z) 1/(aµ) z(0) ~ φ0(z~(0)) 1/(aµ) φλ(z) z~(λ) φλ(~z (λ)) (φ0)2(~z(0))
Forλ = 0, we haveze(0) <1/(aµ) = z(0), because ze(0) <1/(aµ)⇔ µ >
1−α/(a(1−α)) and 1− α/(a(1−α)) < 0. Thus, an equilibrium in the HS-regime at all periods exists for λ = 0 if (41) is satisfied: φ0(ze(0))>0 or
1
µ >1 + α
(1−α)a
An equilibrium experiencing all periods in the T-regime cannot exist, because in afinite number of periods forz0 ≤ez(0)(condition (38)),(φ0)
t
(z0)becomes
negative, which is impossible2.
We assume, however, that(φ0) 2
(ze(0))>0.Thus, the economy can expe-rience an alternation between periods in the HS-regime and periods in the T-regime, and temporary bubbles may exist. Indeed, along such a trajectory, zt shifts between the two values{ze(0),φ0(ez(0))}. As the constraint (41)
re-quires the inequality φ0(zt) > 0 ∀t, it is possible to follow this trajectory only if (φ0)
2
(ez(0))>0.
We wonder if a policy of monetary creation can eliminate the equilibria associated with bubbles, in ensuring the uniqueness of the equilibrium in the HS-regime, such that ∀t, zt=ez(λ).In thefigure, we show how the strait line
2(φ 0) t denotes by definitionφ0◦φ0◦φ0◦...◦φ0 | {z } ttim es .
φλ(z)andez(λ)are shifted whenλ increases. Ifλ is high enough,(φλ)2(ze(λ))
becomes negative, and the economy cannot experience any jump into the T-regime. Thus, bubbles are ruled out.
Formally, it is sufficient to prove the two following points in order to establish this result:
• ∃ λ > 0 such that ze(λ) ≤z(λ) and (φλ) (ze(λ))> 0. Indeed, from (35) and (41), when these two conditions hold, an equilibrium associated with a rate of money creation λ and experiencing all periods in the HS-regime exists.
• (φλ)2(ze(λ)) < 0 : the transition from the HS-regime to the T-regime during one period (or more) is impossible.
The monetary policy no longer allows for a temporary bubble to exist, and ensures the uniqueness of the equilibrium. Let us show these two points.
Proof:
1. We first prove the inequality ze(λ) ≤ z(λ). We know that ze(0) < z(0). The inequality z(λ)−ez(λ)>0 is equivalent to:
λ(1−aµ)> 1−α
α a(1−µ)−1
Asa(1−α)−α <0,this inequality is always satisfied for allλ>0and µ <1.
2. We jointly consider the two inequalities:
(φλ) (ze(λ))>0
(φλ)2(ze(λ))<0
As φλ is an affine function, it is easy to calculate
(φλ)t(ez(λ)) =zb(λ) + (ze(λ)−zb(λ)) µ (1−α)a α(1 +λ) ¶t with b z(λ)≡ − 1−α α(1+λ) ¡ 1− 1+aλλ¢ 1− (1α(1+−α)λa)
Thus, the inequality (φλ)t(ze(λ)) > 0 is equivalent to (after some cal-culations): 1 µ >1 + ¡ 1− aλ 1+λ ¢·³α(1+λ) a(1−α) ´t −1 ¸ 1− (1α(1+−αλ)a) ≡ 1 µt(λ) ⇔µ < µt(λ) (42)
As by assumption α> a(1−α), forλ>0,α(1 +λ)> a(1−α). Thus, for λ given, µt(λ) is a decreasing sequence. And for a given value of t, µt(λ) is a decreasing function of λ. We have to choose λ such that µ < µ1(λ) and µ > µ2(λ), given that µ < µ2(0) (and µ < µ1(0)), with: 1 µ1(λ) = 1 + α[1 + (1−a)λ] a(1−α) (43) 1 µ2(λ) = 1 + α[1 + (1−a)λ] a(1−α) µ 1 + α(1 +λ) a(1−α) ¶ (44) A simple figure allows to find the appropriate values of λ. λ must be in the following interval : λ2 <λ<λ1 with
λ1 = 1 1−a ·µ 1 µ−1 ¶ a1−α α −1 ¸ λ2 = − a2+ p a2 2−4a3a1 2a1 with a1 = (1−a) · α a(1−α) ¸2 a2 = α a(1−α) · 1−a+(2−a)α a(1−α) ¸ a3 = 1− 1 µ+ α a(1−α) µ 1 + α a(1−α) ¶
λ 0 1/µ1(λ) 1/µ2(λ) 1/µ Appropriate values ofλ λ2 λ1
Finally, we have shown that monetary policy makes it possible to obtain a unique equilibrium in the HS-regime. Temporary bubbles can no longer exist. The monetary policy reduces the return of money, and then it becomes impossible for money to have the same return as capital. This result has been obtained under the most unfavorable circumstances: those where the created money is given to young agents. This transfer plays in favor of savings, and tends to diminish the returns on capital. We have shown that this indirect effect is dominated by the direct effect of the monetary policy.
From this result, monetary policy can be viewed as a stabilization in-strument. A temporary bubble on money leads to a deflation period, where money and capital have the same return. It creates a drop in capital accumu-lation and production because it absorbs a share of savings. We have shown that money creation can eliminate such deflation periods as it precludes the appearance of bubbles. This result has a Keynesian flavour, as it is proved that monetary creation can stabilize output fluctuations. But this effect is not provide by demand enhancing, but by bubbles elimination.
5.2
Why
fi
ghting bubbles ?
We have assumed in this section that the objective of the government was to
fight bubbles. Indeed, bubbles reduce capital accumulation in an economy that is experiencing under-accumulation and create indetermination. Such
a government’s objective, however, is partly ad-hoc, because it is not pos-sible to prove that a counter-bubble policy is Pareto improving. We know that the existence of a bubble at some period is beneficial for the genera-tion living during that period, because it increases returns on savings. It is detrimental, however, for the following generations, because it reduces cap-ital accumulation in an economy that is experiencing under-accumulation. Thus, suppressing bubbles cannot be Pareto improving. The counter-bubble monetary policy also has a negative impact on agents’ welfare, because it increases the difference between the returns on the two assets — money and capital — thus increasing the distortion related to money holding.
Nevertheless, it is possible to prove that eliminating temporary bubbles can be welfare improving in a weaker sense. Let us assume that agents who meet at the beginning of time do not know when they will be alive. We can prove that it exists a large range of basic parameters(a,α, µ)such that these agents choose to eliminate bubbles, by setting a high rate of growth of the money supply3.
More precisely we prove this property in two special cases. Indeed, when temporary bubbles may exist, there exists an infinity of equilibria that the economy can experience. Thus, we choose among these equilibria two ex-amples of trajectories that we compare with the unique equilibrium of an economy where an appropriate policy has eliminated bubbles.
Thefirst case is concerned with the long run of the economy. We compare the long run properties of two economies. In the first one, bubbles are ruled out by an appropriate monetary policy, and the economy experiences a con-stant trajectory. In the second one, without monetary policy, it experiences a limit 2-period cycle between HS and T regimes. We prove that long run utility levels are greater in the first economy for a large range of parameters
(a,α, µ). More precisely, it is possible to prove that, whatever their choice criterion is, agents at the beginning of time will prefer to eliminate bubbles for a large range of parameters, as utility levels are greater during an infinite number of periods.
The second argument is more concerned with the short run dynamics. We again compare two economies, starting from the same initial conditions. In the first one, an appropriate monetary policy has eliminated bubbles. The second one without monetary policy experiences at some date one period in the T-regime. We prove that for any number of periods P, there exists parameters such that utility levels in the second economy are smaller than the minimum level in the first economy during at least P periods. Thus, 3This particular notion of welfare improvement is a suggestion af an anonymous referee
that we thank for this idea.
under the veil of ignorance, if agents use a Rawlsian criterion between the different periods, they will prefer to live in the first economy.
5.2.1 The long run argument
We compare the long run properties of two economies. In the first one, bubbles are ruled out by an appropriate monetary policy, and the economy experiences a constant trajectory. In the second one, without monetary pol-icy, it experiences a limit 2-period cycle between HS and T regimes.
For thefirst economy, the long run value of the capital stock per young agents is given by:
k∗HS(λ) = · α 1 +n (1 +λ)φλ(ze(λ)) e z(λ) ¸ 1 1−α
The second economy follows a cycle of period 2. zt oscillates between e
z(0) andφ0(ze(0)). From the preceding part, we now that such trajectory is
possible under the condition (φ0)2(ez(0)). Following (39), kt oscillates in the long run between the 2 values (k1, k2)such that:
(1 +n)k2 = αφ0(ez(0)) e z(0) k α 1 (1 +n)k1 = α(φ0)2(ez(0)) φ0(ze(0)) kα2 Therefore, we obtain: k1 = · α 1 +n[φ0(ez(0))] −1+α 1+α [ez(0)]− α 1+α £(φ 0) 2 (ze(0))¤ 1 1+α ¸ 1 1−α k2 = · α 1 +n[φ0(ez(0))] 1−α 1+α [ez(0)]− 1 1+α £(φ 0) 2 (ez(0))¤ α 1+α ¸ 1 1−α
It remains to prove that long run utility levels are greater in the first economy for some range of parameters (a,α, µ). This property is proved in considering the limit case when µ → µ2(0), with µ < µ2(0). In this case, the value of λ allowing to eliminate bubbles tends toward 0. And, by the definitionµ2(0), we have that (φ0)
2
(ez(0)) tends toward0. Thus, we obtain:
kHS∗ (λ)→ λ→0 · α 1 +n φ0(ze(0)) e z(0) ¸ 1 1−α
and (k1, k2) tends toward (0,0). In the first economy, the long run utility
level tends to some constant value, when in the second economy it tends
to −∞4. In this limit case, an agent who has to choose between these two
long run states will always prefer to live in thefirst economy, whatever is his choice criterion. By continuity, for some choice criterion, an agent always chooses to live in the first economy forµ close toµ2(0).
In this example, we choose for the second economy a case that can be viewed as the worst: bubbles appears all two periods. In the following one, we consider the case of an economy experiencing only one period in the T-regime.
5.2.2 The short run argument
We now consider the dynamics of two economies, starting from the same initial value of capital k0. In the first one, bubbles are ruled out by an
ap-propriate monetary policy λ, and we have for all t zt =ze(λ). From (39), kt follows the following dynamics:
kt1+1 = α 1 +n ¡ kt1¢α · (1 +λ)φλ(ez(λ)) e z(λ) ¸
We assume that the second economy, without monetary policy, experi-ences all periods in the HS-regime except one: between τ and τ + 1, the T-regime occurs. The dynamics of kt are:
∀t 6= τ + 1, kt2+1 = α 1 +n ¡ kt2¢α · φ0(ez(0)) e z(0) ¸ For t = τ + 1, kτ2+2 = α 1 +n ¡ k2t¢α " (φ0)2(ez(0)) φ0(ze(0)) #
When the economy experiences a period in the T-regime betweenτ andτ+1, the real value of money inτ+1is higher and it absorbs higher part of savings. Thus, there is a drop of the level of capital per young in τ + 2.
As in the preceding example, we study the limit case when µ → µ2(0),
with µ < µ2(0). In this case, the value of λ allowing to eliminate bubbles tends toward 0,and(φ0)
2
(ze(0)) tends toward0.Thus, we obtain:
∀t ≤ τ + 1, limkt1 µ→µ2(0) =limkt2 µ→µ2(0) limkτ2+2 µ→µ2(0) = 0
4This last point is easy to prove, as consumption levels(c
1, c2, d1, d2)are respectively bounded by(f(k1), f(k2),(1 +n)f(k1),(1 +n)f(k2)).
If we consider a given integer P, for all values of t such that τ + 2 ≤ t ≤ τ + 2 +P, limk2
t µ→µ2(0)
= 0. By continuity, for a given number P of periods, it is possible to find a range for the parameter µ (in a neighborhood of µ2(0)) and a corresponding value for the monetary policyλ,such that the utility in the second economy is smaller than the minimum value of utility in the first economy during P periods. Thus, under the veil of ignorance, if agents use a Rawlsian criterion between the different periods, they will prefer to live in the first economy.
6
Conclusion
In this paper, we have studied the dynamic properties of an overlapping generations model with capital and money. The medium-of-exchange role of money was taken into account by assuming that agents are subject to a cash-in-advance constraint. We have studied the intertemporal equilibrium of this economy in a general case, without excluding a priori the existence of bubbles (temporary or permanent).
In assuming that a stationary equilibrium exists in an economy without money, we have shown that a monetary equilibrium with a binding liquidity constraint for a sufficiently low liquidity constraint exists within a neighbor-hood of this equilibrium. This monetary equilibrium is not unique, however: we have proved that temporary bubbles on money may appear at some pe-riods. Notably, we have shown that, within the neighborhood of an equi-librium without bubbles, cyclical equilibria exist along which the economy experiences n periods of temporary bubbles and p periods without bubbles, nandpbeing some integers. Thus, the existence of temporary bubbles leads to a multiplicity of equilibria.
Finally, we have assumed that the government’s objective is tofight bub-bles using a monetary policy. Assuming Cobb-Douglas functions for utility and production, we have shown that a high enough rate of monetary creation eliminates temporary bubbles. Thus, the economy follows the only equilib-rium trajectory that does not have bubbles, and indeterminacy disappears. We develop two arguments in favor of such a policy. In the long run, we have shown that the periodic appearance of bubbles can lead to a utility level smaller than the one reached along a bubbleless trajectory. In the short run, we have shown that the only appearance of a one period bubble is enough to drop the utility levels during many periods far from the level reached along the bubbleless trajectory. For these reasons, fighting bubbles can be an objective of monetary policy.
Appendix 1: proof of proposition 1.
A trajectory of the HS-regime is characterized by the dynamical equation given by (29):
g(kt+1, kt, µ,λ) = 0, t≥0
The condition that assures that money is a dominated asset (24), with (22), is equivalent to:
(1 +n)kt+1 < R(kt)kt(1 +λ)
By assumption, we have g(kD, kD,0,λ) = 0. g being continuously diff eren-tiable with respect to the first partial derivative g10(kD, kD,0,λ)6= 0, we can
apply the implicit function theorem. There existsε>0andI neighborhood of kD,such that for allk
∈I and allµ,|µ|<ε,the equationg(x, k, µ,λ) = 0
admits a unique solution x = h(k, µ) in a neighborhood of kD5. We have: kD =h(kD,0)and(1 +n)kD< R(kD)kD(1 +λ).It is possible to restrict the neighborhoods in order that the solution satisfies: (1 +n)x < R(k)k(1 +λ), for all k∈I and all µ,|µ|<ε.
The functionh is differentiable and satisfies: h01(kD,0) =−g02/g10 = σ
0
ww0(kD)
1 +n−σ0
RR0(kD)
This derivative is strictly between 0 and 1. We can again restrict the neigh-borhoods so that the derivative h01(k, µ) is also between 0 and 1. Applying
the implicit function theorem to the two-variable function g(k, k, µ,λ), we obtain a stationary equilibrium k∗(µ)∈I for allµ < ε.
Let us show that for allk0 ∈I,the trajectory kt(µ) such that: kt+1(µ) =h(kt(µ), µ), k0(µ) =k0
is defined, belongs to I and converges towardsk∗(µ). By recurrence, ifkt(µ)∈I,
kt+1(µ)−k∗(µ) = h(kt(µ), µ)−h(k∗(µ), µ)
= h01(xt(µ), µ) (kt(µ)−k∗(µ))
with xt(µ) between k∗(µ) and kt(µ) and belonging to I. As h01 ∈ (0,1), the
distance between kt+1(µ) and k∗(µ) is strictly smaller than the one between
kt(µ)andk∗(µ). Thus,kt+1(µ)is defined and belongs toI,and the sequence
kt(µ)−k∗(µ) tends toward 0.
5Asλis assumed to befixed in this part, it is not useful to make explicit the dependence
ofxwith respect toλ.
The sequence kt(µ) belongs to the HS-regime as it satisfies:
(1 +n)kt+1(µ)< R(kt(µ))kt(µ)(1 +λ) from the preceding restrictions taken on I.
The uniform convergence of this sequence when µ → 0 results from its convergence and the convergence of k∗(µ) towards k∗(0) = kD. Indeed, we can write: |kt(µ)−kt(0)|≤|kt(µ)−k∗(µ)|+ ¯ ¯k∗(µ)−kD¯¯+¯¯kt(0)−kD ¯ ¯
From the preceding recurrence, the first and third terms can be bounded above by a sequence zt ≥0, which has a null limit and does not depend on µ. Thus, ∀ε > 0, when t is high enough ( t ≥ T), these two terms can be made smaller thanε.We also have: lim
µ→0
¯
¯k∗(µ)−kD¯¯= 0. Finally, for t < T, the continuity with respect to the variableµ makes it possible to obtain the convergence: sup
t<T |
kt(µ)−kt(0)|→0 whenµ→0.
Appendix 2: proof of proposition 3.
We define the following notation:
χ(k, m, µ) = (1 +n)R(k)k (1−µ)(m+ (1 +n)k)
The cyclical orbit of period n+p · µ m1 k1 ¶ µ m2 k2 ¶ ... µ mn kn ¶ µ mn+1 kn+1 ¶ µ mn+2 kn+2 ¶ ... µ mn+p kn+p ¶ ¸
has to satisfy the following equations:
Between periods n+p and1 : HS-regime m1−µ(1+1−µλ)R(k1)k1 = 0 (1 +n)k1−σ £ w(kn+p) +1+λλmn+p,χ(k1, mn+p, µ) ¤ +mn+p = 0 Between periods 1and2 : T-regime
m2−(1+1+λ)Rn(k2)m1 = 0 (1 +n)k2−σ £ w(k1) +1+λλm1, R(k2) ¤ +m1 = 0
Between periods 2and3 : T-regime m3−(1+1+λ)Rn(k3)m2 = 0 (1 +n)k3−σ £ w(k2) +1+λλm2, R(k3) ¤ +m2 = 0 .. .
Between periods nandn+ 1 : T-regime mn+1− (1+λ1+)R(nkn+1)mn = 0 (1 +n)kn+1−σ £ w(kn) +1+λλmn, R(kn+1) ¤ +mn= 0 Between periods n+ 1 and n+ 2 : HS-regime
mn+2− µ(1+1−µλ)R(kn+2)kn+2 = 0 (1 +n)kn+2−σ £ w(kn+1) +1+λλmn+1,χ(kn+2, mn+1, µ) ¤ +mn+1 = 0
Between periods n+ 2 and n+ 3 : HS-regime mn+3− µ(1+1−µλ)R(kn+3)kn+3 = 0 (1 +n)kn+3−σ £ w(kn+2) +1+λλmn+2,χ(kn+3, mn+2, µ) ¤ +mn+2 = 0 .. .
Between periods n+p−1andn+p: HS-regime mn+p− µ(1+λ) 1−µ R(kn+p)kn+p = 0 (1 +n)kn+p −σ £ w(kn+p−1) +1+λλmn+p−1,χ(kn+p, mn+p−1, µ) ¤ +mn+p−1 = 0 (45) In addition to these equations, the periodic equilibrium has to satisfy n+pconstraints that have the same expression in both types of regimes (cf. constraints (24) and (27)):
∀t= 1, ....n+p, (1−µ)mt> µ(1 +n)kt+1 (46)
(with the convention that kn+p+1 =k1).
We first show the existence of a unique solution to the system (45) for each value of µ within a neighborhood of 0. We then prove that such a solution satisfies all the constraints (46) whenµ is sufficiently small.
All conditions in the system (45) are satisfied for µ = 0, ki = kD and mi = 0. We denote the vector of dimension 2(n+p) byX
X = (m1, m2, ....mn, mn+1, ...mn+p, k1, k2, ...kn, kn+1, ...kn+p) and we note: X0 = 0,0, ....0,0, ...0 | {z } n+ptimes , kD, kD, ...kD, kD, ...kD | {z } n+ptimes
The2(n+p) preceding equations can be expressed by defining function Z :R2(n+p)+1 →R2(n+p), such that: Z(X, µ) = 0 with: Z(X, µ)≡ m1− µ(1+1−µλ)R(k1)k1 m2− (1+λ1+)Rn(k2)m1 m3− (1+λ1+)Rn(k3)m2 .. . mn+1− (1+λ)1+R(nkn+1)mn mn+2− µ(1+1−µλ)R(kn+2)kn+2 mn+3− µ(1+1−µλ)R(kn+3)kn+3 .. . mn+p− µ(1+1−µλ)R(kn+p)kn+p (1 +n)k2 −σ £ w(k1) +1+λλm1, R(k2) ¤ +m1 (1 +n)k3 −σ £ w(k2) +1+λλm2, R(k3) ¤ +m2 .. . (1 +n)kn+1−σ £ w(kn) +1+λλmn, R(kn+1) ¤ +mn (1 +n)kn+2−σ £ w(kn+1) +1+λλmn+1,χ(kn+2, mn+1, µ) ¤ +mn+1 (1 +n)kn+3−σ £ w(kn+2) +1+λλmn+2,χ(kn+3, mn+2, µ) ¤ +mn+2 .. . (1 +n)kn+p−σ £ w(kn+p−1) +1+λλmn+p−1,χ(kn+p, mn+p−1, µ) ¤ +mn+p−1 (1 +n)k1 −σ £ w(kn+p) + 1+λλmn+p,χ(k1, mn+p, µ) ¤ +mn+p
We have chosen this particular order for the different equations, because it makes easier the calculations. We know that Z(X0,0) = 0 and that Z is
continuously differentiable within a neighborhood of(X0,0).If the differential
dXZ(X0,0) is bijective from R2(n+p) on R2(n+p), the equation Z(X, µ) = 0
defines an implicit function within a neighborhood of µ= 0.More precisely, ∃α > 0 and β > 0, such that ∀µ, 0 ≤ µ < α, ∃!X ∈ B(X0,β), such that:
Z(X, µ) = 0.
Consequently, it remains to be proved that dXZ(X0,0) is bijective. We
set: A = −σ0w£w¡kD¢, R¡kD¢¤w0(kD) B = 1 +n−σ0R£w¡kD¢, R¡kD¢¤R0(kD) C = 1 +σ 0 R £ w¡kD¢, R¡kD¢¤R(kD) (1 +n)kD − λ 1 +λσ 0 w £ w¡kD¢, R¡kD¢¤ D = 1− λ 1 +λσ 0 w £ w¡kD¢, R¡kD¢¤
In the calculus ofdXZ(X0,0), it is important to note that:
∂χ ∂k(k D,0,0) = R0(kD) ∂χ ∂m(k D,0,0) = − R(k D) (1 +n)kD
We define Ij,j as the identity matrix of dimension (j, j). We also define Nj,jh andNl
j,j as the matrixes of dimension(j, j) such that:
Nj,jh = 0 1 0 0 0 1 . .. ... . .. 1 0 0 Nj,jl =¡Nj,jh ¢0 = 0 0 0 1 0 0 1 . .. . .. 0 0 0 1 0
Thus,dXZ(X0,0)can be written:
dXZ(X0,0) = En,n 0n,p 0n,n 0n,p Fp,n Ip,p 0p,n 0p,p DIn,n 0n,p Gn,n Hn,p 0p,n CIp,p Hp,n Jp,p
where: En,n =In,n− (1 +λ)R(kD) 1 +n N l n,n Fp,n= 0 ... 0 −(1+1+λ)Rn(kD) 0 ... 0 0 .. . ... ... ... 0 ... 0 0 Gn,n =AIn,n+BNn,nh Hi,j = 0 0 ... 0 0 ... 0 .. . ... ... B 0 ... 0 Jp,p =AIp,p+BNp,ph
Developing the determinant of this matrix by the lines 1, ...n+p, we
find: detdXZ(X0,0) = det · Gn,n Hn,p Hp,n Jp,p ¸
Then, developing this last determinant by the first column, we have:
detdXZ(X0,0) = An+p+ (−1)1+n+pBn+p =An+p−(−B)
n+p
We know that A < 0. From assumption 2, we know that B > 0 and that A +B > 0 or A > −B. Thus, detdXZ(X0,0) 6= 0 and has the sign of
(−1)n+p+1.
The second part of the demonstration needs to prove that (46) holds when µ is small enough.
Fort, such that n+ 2 ≤t≤n+p ort= 1, mt is such that:
mt= µ(1 +λ) 1−µ R(kt)kt Replacing mt in (46) gives: ∀t s. t. n+ 2≤t≤n+p ort= 1, (1 +λ)R(kt) 1 +n kt > kt+1 (47) Fort, such that 2≤t ≤n+ 1,it is easy with (45) to obtain:
mt = (1 +λ)R(kt) 1 +n (1 +λ)R(kt−1) 1 +n ... (1 +λ)R(k2) 1 +n µ(1 +λ) 1−µ R(k1)k1
Replacing mt in (46) gives,∀ t such that 2≤t≤n+ 1, (1 +λ)R(kt) 1 +n (1 +λ)R(kt−1) 1 +n ... (1 +λ)R(k2) 1 +n (1 +λ)R(k1) 1 +n k1 > kt+1 (48) Thus, we have proved that the system (45) with all the constraints (46) is equivalent to the system (45) with (47) for n+ 2≤t≤n+por t= 1and (48) for 2≤t≤n+ 1.
Finally, from assumption 2, (47) and (48) are satisfied in X0, because
R(kD)> (1 +n)/(1 +λ). Thus, (47) and (48) are satisfied when µ is small enough.
Appendix 3: the intertemporal equilibrium in a Cobb-Douglas economy
With a Cobb-Douglas utility function, the utility maximization under the budgetary constraints gives the same expression of total savings in both regimes:
σt=a(wt+θt) =st+mt (49)
Indeed, savings no longer depend on their return.
As total savingsσtis the same in the two regimes, it is possible to express (23) and (26) by the same equation:
(1 +n)kt+1 =awt− µ 1− aλt 1 +λt ¶ mt (50)
Finally, we write the specific conditions in each regime. We introduce the new variable
zt = wt mt
HS-regime:
Real balances of the agents, given by (22), now become:
(1−µ)mt+1 = (1 +λt+1)µαktα+1 = (1 +λt+1)µ
α
1−αwt+1 (51)
thus, (51) can be written: zt+1 = 1−µ µ 1−α α 1 1 +λt+1 ≡ e z(λt+1) (52)
The condition ensuring that the return on money does not exceed the one on capital (24) was:
µ(1 +n)kt+1 ≤(1−µ)mt
In using (50) for eliminating the variable kt+1, this condition becomes: zt ≤ 1 a µ 1 µ − aλt 1 +λt ¶ (53) The T-regime:
In using (50) and the variable zt, equation (25) can be written: zt+1 = 1−α α(1 +λt+1) · azt− µ 1− aλt 1 +λt ¶¸ (54) The condition ensuring that the liquidity constraint is satisfied remains:
zt < 1 a µ 1 µ− aλt 1 +λt ¶ (55) Finally, the liquidity constraint for thefirst old agents (28) can be written:
(1−µ) m0 1 +λ0 ≥ µR0k0 =µ α 1−αw0 or: z0 ≤ 1 1 +λ0 1−µ µ 1−α α =ez(λ0) (56)
Finally, from (50), in both regimes the corresponding dynamics ofkt can be written: (1 +n)kt+1 = wt zt · azt− µ 1− aλt 1 +λt ¶¸
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